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Singular Chains on Topological Stacks

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Abstract

The main objective of this thesis is to introduce the concept of `singular chains on topological
stacks'. The idea is to functorially associate to a topological stack, a simplicial set which captures
its homotopy type.
This will allow us to compute the singular homology and cohomology of topological stacks.
Noohi and Behrend have given several approaches to this problem, however all of these approaches
rely on the choice of an atlas for a topological stack. We shall show that our new approach agrees
with the existing approaches but has the advantage of being functorial.
Noohi has introduced weak equivalences and brations of topological stacks. In analogy to the
singular chains functor for topological spaces, we shall show that the functor Sing preserves the
weak equivalences and brations de ned by Noohi under certain ` brancy conditions'.
In the second part, we shall push the analogy with the topological singular chains further by
considering the adjunction with the geometric realization and the associated counit. We develop
a corresponding (but weaker) notion for topological stacks.
We shall give a method for computing the homotopy type of a stack which has a groupoid
presentation. Finally, we shall compute the homotopy type of certain mapping stacks and develop
the totalization of a cosimplicial topological stack. We shall indicate how this (using the approach
of Cohen and Jones) gives a method for computing the string topology of a topological stack.